In this work, we start exploring and presenting the design of
numerical approximations of the MHD equations based on the Virtual
Element Method (VEM).
Our goal is to address at least part of the issues discussed so far,
but in a framework that makes it possible to address all of them in
the future developments of this research project.
The VEM, initially proposed
in [1]} for
the numerical approximation of elliptic problems, is a Galerkin
projection method such as the Finite Element Method (FEM).
The major difference between the VEM and the FEM is that in the formulation
and practical implementation of the VEM we do not need an explicit
knowledge of the basis functions that generate the finite element
approximation spaces.
In fact, all the bilinear forms and linear functionals of the discrete
variational formulation are built using suitable polynomial
projections that are always computable from a careful choice of the
degrees of freedom.
For these reasons, such approximation spaces and the method itself are
dubbed as “virtual”.
Local polynomial consistency and an additional stability term provide
the well-posedness of the final discretization.
Since the formulation of the method does not need a closed form of the
basis functions, the resulting computational framework is extremely
powerful and offers important advantages with respect to the FEM.
We can build approximation spaces that are in principle of any order
of accuracy and global regularity, suitable to general polyhedral meshes,
even including nonconvex or nonconforming elements in 2D and 3D, and,
importantly, satifying other additional properties such as being part
of a discrete de Rham chain.
This last property has the major consequence that both the numerical
approximation of the fluid velocity and the magnetic flux field are
intrinsically divergence free, a fact that we can verify numerically
up to the machine precision.
